Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0.

Question

A. 0
B. 1
C. 2
D. 3
E. 4

freedom128 · Accepted Answer

x^2 -2|x| -8 =0

If x is 0 or positive, then x^2 -2x -8 =0
--> (x-4)(x+2)=0
--> x=4 (Yes) or x=-2 (No!)

If x is negative, then x^2 +2x -8 =0
--> (x+4)(x-2)=0
--> x=-4 (Yes) or x=2 (No!)

There are only two solutions:
x=-4 and x=4

Answer is (C)

anuradha619 · Answer

Hey, why are -2 and +2 not considered a solution for the problem?

Bunuel · Answer

Because -2 and 2 do not satisfy x^2 - 2|x| - 8 = 0.

If x = -2, then x^2 - 2|x| - 8 = 4 - 4 - 8 = -8, not 0.
If x = 2, then x^2 - 2|x| - 8 = 4 - 4 - 8 = -8, not 0.

UncleFrodo · Answer

because they are eliminated by conditions that we apply to find solutions and to open modulus. if we have |x| we can open modulus if we set next conditions (https://en.wikipedia.org/wiki/Absolute_value): 1. x>=0 ... 2. x<0 ... and this conditions are stated in the solution above If x is 0 or positive, then x^2 -2x -8 =0 --> (x-4)(x+2)=0 --> x=4 (Yes) or x=-2 (No!) - because x is 0 or positive If x is negative , then x^2 +2x -8 =0 --> (x+4)(x-2)=0 --> x=-4 (Yes) or x=2 (No!) because x is negative

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

We need to find the number of real solutions of \(x^2 - 2|x| - 8 = 0\)

As we have |x| in the equation so we will have two cases

As we have \|x\| in the equation so we will have two cases
-Case 1: x ≥ 0 => \|x\| = x => \(x^2 - 2x - 8 = 0\) => \(x^2 - 4x + 2x - 8 = 0\) => x(x - 4) + 2(x - 4) = 0 => (x - 4) * (x + 2) = 0 => x = 4, -2 But condition was x ≥ 0 => x = 4 is a SOLUTION	-Case 2: x < 0 => \|x\| = -x => \(x^2 - 2(-x) - 8 = 0\) => \(x^2 + 2x - 8 = 0\) => \(x^2 + 4x - 2x - 8 = 0\) => x(x + 4) - 2(x + 4) = 0 => (x + 4) * (x - 2) = 0 => x = -4, 2 But condition was x < 0 => x = -4 is a SOLUTION

=> There are 2 real solutions possible

So, Answer will be C
Hope it helps!

Watch the following video to MASTER Absolute Values

Apeksha2101 · Answer

But when we take x=2 in equation 2 we see it satisfies the equation.

Bunuel · Answer

No, it does not.

If x = 2, then x^2 - 2|x| - 8 = 4 - 4 - 8 = -8, not 0.

bumpbot · Answer

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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0.

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Find the number of real solutions of the equation x^2 - 2|x| - 8 = 0.

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